Use of JMP (Release 3) for plotting Weibull censored data and estimating
parameters will be illustrated using data from a previous example.

Steps in a Weibull analysis using JMP software

Weibull Data Example

Failure times were 55, 187, 216, 240, 244, 335, 361, 373, 375, and 386
hours, and 10 unfailed units were removed from test at 500 hours. The steps
in creating a JMP worksheet and analyzing the data are as follows:

1. Set up three columns, one for the failure and censoring times ("Time"),
another to indicate whether the time is a failure or a censoring time ("Cens")
and the third column to show how many units failed or were censored at
that time ("Freq"). Fill in the 11 times above, using "0" in Cens to indicate
a failure and "1" in Cens to indicate a censoring time. The spreadsheet
will look as follows:

You can obtain a copy of this JMP worksheet by clicking here
mleex.jmp
. If your browser is configured to bring up JMP automatically, you can
try out the example as you read about it.

2. Click on Analyze, choose "Survival" and then choose "Kaplan - Meier
Method". Note: Some software packages (and other releases of JMP)
might use the name "Product Limit Method" or "Product Limit Survival Estimates"
instead of the equivalent name "Kaplan-Meier".

3. In the box that appears, select the columns from mleex that
correspond to "Time", "Censor" and "Freq", put them in the corresponding
slots on the right (see below) and click "OK".

4. Click "OK" and the analysis results appear. You may have to use the
"check mark" tab on the lower left to select Weibull Plot (other choices
are Lognormal and Exponential). You may also have to open the tab next
to the words "Weibull Plot" and select "Weibull Estimates". The results
are shown below.

Note: JMP uses the parameter
for the Weibull characteristic life (as does Dataplot), and the parameter
for the shape (Dataplot uses ).
The Extreme Value distribution parameter estimates are for the distribution
of "ln time to fail" and have the relationship

5. There is an alternate way to obtain some of the same results, which
can also be used to fit models when there are additional "effects" such
as temperature differences or vintage or plant of manufacturing differences.
Instead of clicking "Kaplan - Meier Method" in step 2, chose "Parametric
Model" after selecting "Survival" from the "Analysis" choices. The screen
below appears. Repeat step 3 and make sure "Weibull" appears as the "Get
Model" choice. In this example there are no other effects to "Add" (the
acceleration
model example later on will illustrate how to add a temperature effect).
Click "Run Model" to obtain the results below. This time, you need to use
the check symbol tab to obtain confidence limits. Only the Extreme Value
distribution parameter estimates are displayed.

Limitations and
a warning about the Likelihood calculation in JMP

Notes:

1. The built in reliability analysis routine that iscurrently part of
JMP only handles exact time of failure data with possible right censoring.
However, the use of templates (provided later in the Handbook) for either
Weibull or lognormal data extends JMP analysis capabilities to handle readout
(interval) data and any type of censoring or truncation. This will be described
in the acceleration
model example later on.

2. The "Model Fit" screen for the Weibull model gives a value for -Loglikelihood
for the Weibull fit. This should be the negative of the maximized likelihood
function. However, JMP leaves out a term consisting of the sum of all the
natural logarithms of the times of failures in the data set. This does
not affect the calculation of MLE's or confidence bounds but can be confusing
when comparing results between different software packages. In the example
above, the sum of the ln times is ln 55 + ln 187 + . . . + ln 386 = 55.099
and the correct maximum log likelihood is - (20.023 + 55.099) = - 75.122.

3. The omission of the sum of the ln times of failures in the likelihood
also occurs when fitting lognormal and exponential models.

MLE analysis is an accurate and easy way to estimate life distribution
parameters, provided that a good software analysis package is available.
The package should also calculate confidence bounds and loglikelihood values.
JMP has this capability, as do several other commercial statistical analysis
packages.